Environmental Engineering Reference

In-Depth Information

Figure 5.10. Dynamics described by model (
5.17
) when
a
is negative (
a

=−

0

.

1,

D

=

10,
k
0
=

1). The three panels correspond to
t
equal to 0, 5, and 35 time units, the

field has 128

128 pixels, and periodic boundary conditions are set. The gray-tone

scale spans the interval [10
−
3

×

10
3
]. The initial condition is

,

φ

(
r

,

t

=

0)

=
ξ
u
, where

ξ
u
is a noise uniformly distributed in the interval [10
−
2

10
2
].

,

tend to vanish as the system tends to the homogeneous stable state

φ
=

0. Notice that

the transient pattern exhibits the periodicity of about 2

π/

k
0
pixels imposed by the

spatial coupling (see Appendix B).

Conversely, when
a
is positive no steady states exist and the dynamics of

diverge.

However, even in this case the spatial terms in (
5.17
) are able to induce patterns

that become more and more pronounced (i.e., with stronger spatial gradients) as the

dynamics diverge (see Fig.
5.11
). To stabilize the dynamics to a steady state, it is

necessary to introduce a nonlinear term that hampers the dynamics in diverging.

A relatively simple nonlinear term is

φ

3
; in this case the dynamics turn into the

celebrated Ginzburg-Landau model, with local dynamics expressed as
f
(

−
φ

φ

)

=

a

φ
−

3
,

φ

∂φ

∂

3

2

k
0
)
2

t
=

a

φ
−
φ

−

D
(

∇

+

φ.

(5.18)

−
√
a

,
√
a
] - the diverging effect of the lin-

For small values of

φ

- in the interval [

ear term
a

φ

prevails (with
a

>

0), and as

φ

increases the stabilizing effect of the

Figure 5.11. Example of the dynamics described by deterministic model (
5.17
) when

a
is positive (
a

=+

.

=

10,
k
0
=

1). The three panels correspond to
t
equal to

0, 30, and 60 time units, and the gray-tone scale spans the interval [

0

1,
D

−

.

,

.

0

1

0

1]. The

other conditions are as in Fig.
5.10
.

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